A New Perspective on the Average Mixing Matrix

TL;DR

This paper explores the average mixing matrix of continuous-time quantum walks on graphs, revealing its mathematical structure and connections to graph automorphisms, offering new insights into quantum walk behavior.

Contribution

It introduces a novel formulation of the average mixing matrix using the commutant algebra, linking its rank to graph automorphisms.

Findings

The average mixing matrix is the transformation of the orthogonal projection onto the commutant algebra.

Connections are established between the rank of the average mixing matrix and graph automorphisms.

The formulation provides a new perspective on quantum walk mixing properties.

Abstract

We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.